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qpOASES: a parametric active-set algorithm for quadratic programming. (English) Zbl 1302.90146
Summary: Many practical applications lead to optimization problems that can either be stated as quadratic programming (QP) problems or require the solution of QP problems on a lower algorithmic level. One relatively recent approach to solve QP problems are parametric active-set methods that are based on tracing the solution along a linear homotopy between a QP problem with known solution and the QP problem to be solved. This approach seems to make them particularly suited for applications where a-priori information can be used to speed-up the QP solution or where high solution accuracy is required. In this paper we describe the open-source C++ software package qpOASES, which implements a parametric active-set method in a reliable and efficient way. Numerical tests show that qpOASES can outperform other popular academic and commercial QP solvers on small- to medium-scale convex test examples of the Maros-Mészáros QP collection. Moreover, various interfaces to third-party software packages make it easy to use, even on embedded computer hardware. Finally, we describe how qpOASES can be used to compute critical points of nonconvex QP problems.

MSC:
90C20 Quadratic programming
65K05 Numerical mathematical programming methods
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[1] Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Croz, J.D., Greenbaum, A., Hammarling, S., McKenney, A., Sorensen, D.: LAPACK Users’ Guide, 3rd edn. SIAM, Philadelphia, PA (1999) · Zbl 0934.65030
[2] Ängeby, J., Huschenbett, M., Alberer, D.: Automotive Model Predictive Control, MIMO Model Predictive Control for Integral Gas Engines. In: Lecture Notes in Control and Information Sciences, vol. 402, pp. 257-272, Springer, Berlin (2010) · Zbl 1222.93067
[3] Arnold, E., Neupert, J., Sawodny, O., Schneider, K.: Trajectory tracking for boom cranes based on nonlinear control and optimal trajectory generation. In: IEEE International Conference on Control Applications, pp. 1444-1449 (2007). doi:10.1109/CCA.2007.4389439 · Zbl 1284.93100
[4] Bartels, R; Golub, G, The simplex method for linear programming using LU decomposition, Commun. ACM, 12, 266-268, (1969) · Zbl 0181.19104
[5] Bartlett, R; Biegler, L, Qpschur: a dual, active set, Schur complement method for large-scale and structured convex quadratic programming algorithm, Optim. Eng., 7, 5-32, (2006) · Zbl 1167.90615
[6] Benzi, M; Golub, G; Liesen, J, Numerical solution of saddle-point problems, Acta Numerica, 14, 1-137, (2005) · Zbl 1115.65034
[7] Best, M.: An Algorithm for the Solution of the Parametric Quadratic Programming Problem. In: Applied Mathematics and Parallel Computing, pp. 57-76. Physica, Heidelberg (1996) · Zbl 0906.65064
[8] Blackford, LS; Demmel, J; Dongarra, J; Duff, I, An updated set of basic linear algebra subprograms (BLAS), ACM Trans. Math. Softw., 28, 135-151, (2002) · Zbl 1070.65520
[9] Bock, H., Plitt, K.: A multiple shooting algorithm for direct solution of optimal control problems. In: Proceedings 9th IFAC World Congress, pp. 242-247. Pergamon Press, Budapest (1984) · Zbl 1167.90615
[10] Contesse, L, Une charactérisation complète des minima locaux en programmation quadratique, Numerische Mathematik, 34, 315-332, (1980) · Zbl 0422.90061
[11] Dantzig, G.B.: Linear Programming and Extensions. Princeton University Press, Princeton (1963) · Zbl 0108.33103
[12] Diehl, M., Ferreau, H.J., Haverbeke, N.: Nonlinear model predictive control, Efficient Numerical Methods for Nonlinear MPC and Moving Horizon Estimation. In: Lecture Notes in Control and Information Sciences, vol. 384, pp. 391-417. Springer, Berlin (2009) · Zbl 1195.93038
[13] Eaton, J.W.: GNU Octave Manual. Network Theory Limited (2002)
[14] Ferreau, H.J.: An Online Active Set Strategy for Fast Solution of Parametric Quadratic Programs with Applications to Predictive Engine Control. Master’s thesis, University of Heidelberg (2006) · Zbl 0537.90081
[15] Ferreau, H.J., et al.: qpOASES User’s Manual. http://www.qpOASES.org (2007-2014) · Zbl 0727.65055
[16] Ferreau, H.J.: Model predictive control algorithms for applications with millisecond timescales. PhD thesis, KU Leuven (2011)
[17] Ferreau, H.J., Diehl, M.: Online QP Benchmark Collection. http://www.qpOASES.org/onlineQP (2006-2008)
[18] Ferreau, HJ; Ortner, P; Langthaler, P; Re, L; Diehl, M, Predictive control of a real-world diesel engine using an extended online active set strategy, Annu. Rev. Control, 31, 293-301, (2007)
[19] Ferreau, HJ; Bock, HG; Diehl, M, An online active set strategy to overcome the limitations of explicit MPC, Int. J. Robust Nonlinear Control, 18, 816-830, (2008) · Zbl 1284.93100
[20] Fletcher, R.: Practical Methods of Optimization, 2nd edn. Wiley, Chichester (1987) · Zbl 0905.65002
[21] Fletcher, R.: Approximation Theory and Optimization, Dense factors of sparse matrices. In: Tributes to M.J.D. Powell, pp. 145-166. Cambridge University Press, Cambridge (1997) · Zbl 1031.65043
[22] Fletcher, R.: Stable reduced Hessian updates for indefinite quadratic programming. Numerical Analysis Report NA/187, Department of Mathematics and Computer Science, University of Dundee, Dundee DD1 4HN, Scotland, UK (1999) · Zbl 0964.65065
[23] Fletcher, R; Matthews, S, Stable modification of explicit LU factors for simplex updates, Math. Progr., 30, 267-284, (1984) · Zbl 0574.65057
[24] Gertz, E; Wright, S, Object-oriented software for quadratic programming, ACM Trans. Math. Softw., 29, 58-81, (2003) · Zbl 1068.90586
[25] Gill, P; Golub, G; Murray, W; Saunders, MA, Methods for modifying matrix factorizations, Math. Comput., 28, 505-535, (1974) · Zbl 0289.65021
[26] Gill, P; Murray, W; Saunders, M; Wright, M, Procedures for optimization problems with a mixture of bounds and general linear constraints, ACM Trans. Math. Softw., 10, 282-298, (1984) · Zbl 0562.90079
[27] Gill, P; Murray, W; Saunders, M; Wright, M, Maintaining LU factors of a general sparse matrix, Linear Algebra Appl., 88, 239-270, (1987) · Zbl 0618.65019
[28] Goldfarb, D; Idnani, A, A numerically stable dual method for solving strictly convex quadratic programs, Math. Progr., 27, 1-33, (1983) · Zbl 0537.90081
[29] Gould, N, An algorithm for large-scale quadratic programming, IMA J. Numer. Anal., 11, 299-324, (1991) · Zbl 0727.65055
[30] Gould, N., Toint, P.: A quadratic programming bibliography. Tech. Rep. 2000-1, Rutherford Appleton Laboratory, Computational Science and Engineering Department (2010) · Zbl 1216.90069
[31] Gould, N., Toint, P.: A quadratic programming page. http://www.numerical.rl.ac.uk/qp/qp.html (2012) · Zbl 1235.90118
[32] Gould, N., Orban, D., Toint, P.: CUTEr testing environment for optimization and linear algebra solvers. http://cuter.rl.ac.uk/cuter-www/, (2002)
[33] Harris, P, Pivot selection methods of the DEVEXLP code, Math. Progr., 5, 1-29, (1973) · Zbl 0261.90031
[34] van Heesch, D.: Doxygen homepage (1997-2011). http://www.doxygen.org · Zbl 0358.90053
[35] Houska, B; Ferreau, HJ; Diehl, M, ACADO toolkit—an open source framework for automatic control and dynamic optimization, Optim. Control Appl. Methods, 32, 298-312, (2011) · Zbl 1218.49002
[36] Huynh, H.: A large-scale quadratic programming solver based on block-LU updates of the KKT system. PhD thesis, Stanford University (2008) · Zbl 0422.90061
[37] IBM Corp: IBM ILOG CPLEX V12.1, User’s Manual for CPLEX (2009)
[38] Inc TM: Real-Time Workshop for Use with SIMULINK, User’s Guide (1999)
[39] Karmarkar, N, A new polynomial time algorithm for linear programming, Combinatorica, 4, 373-395, (1984) · Zbl 0557.90065
[40] Kirches, C; Bock, H; Schlöder, J; Sager, S, A factorization with update procedures for a KKT matrix arising in direct optimal control, Math. Progr. Comput., 3, 319-348, (2011) · Zbl 1276.90046
[41] Kostina, E, The long step rule in the bounded-variable dual simplex method: numerical experiments, Math. Methods Oper. Res., 55, 413-429, (2002) · Zbl 1031.90010
[42] Leineweber, D; Bauer, I; Schäfer, A; Bock, H; Schlöder, J, An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization (parts I and II), Comput. Chem. Eng., 27, 157-174, (2003)
[43] Löfberg, J.: YALMIP: A toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD Conference, Taipei, Taiwan (2004). http://users.isy.liu.se/johanl/yalmip
[44] Maros, I; Meszaros, C, A repository of convex quadratic programming problems, Optim. Methods Softw., 11, 431-449, (1999) · Zbl 0973.90520
[45] Murty, K, Some NP-complete problems in quadratic and nonlinear programming, Math. Progr., 39, 117-129, (1987) · Zbl 0637.90078
[46] Nesterov, Y.: Introductory lectures on convex optimization: a basic course. In: Applied Optimization, vol. 87. Kluwer Academic Publishers, Dordrecht (2003) · Zbl 1086.90045
[47] Nesterov, Y., Nemirovski, A.: Interior-point Polynomial Algorithms in Convex Programming. In: Society for Industrial Mathematics (1994)
[48] Nocedal, J., Wright, S.: Springer series in operations research and financial engineering. In: Numerical Optimization, 2nd edn. Springer, Berlin (2006) · Zbl 0637.90078
[49] Rauter, G., von Zitzewitz, J., Duschau-Wicke, A., Vallery, H., Riener, R.: A tendon-based parallel robot applied to motor learning in sports. In: Proceedings of the IEEE International Conference on Biomedical Robotics and Biomechatronics 2010, Japan (2010)
[50] Rockafellar, R, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14, 877-898, (1976) · Zbl 0358.90053
[51] Sager, S.: Lange Schritte im Dualen Simplex-Algorithmus. Master’s thesis, Universität Heidelberg (2001) · Zbl 0103.37603
[52] Scilab Consortium: Scilab: The free software for numerical computation. Scilab Consortium, Digiteo, Paris, France (2011). http://www.scilab.org · Zbl 0181.19104
[53] Takács, G., Rohal’-Ilkiv, B.: Predictive vibration control: efficient constrained MPC vibration control for lightly damped mechanical systems. Springer, Berlin (2012) · Zbl 1253.93005
[54] Van den Broeck, L.: Time optimal control of mechatronic systems through embedded optimization. PhD thesis, KU Leuven (2011)
[55] Wang, X.: Resolution of Ties in Parametric Quadratic Programming. Master’s thesis, University of Waterloo, Ontario, Canada (2004)
[56] Wilkinson, J.: The Algebraic Eigenvalue Problem. Clarendon Press, Oxford (1965) · Zbl 0258.65037
[57] Wills, A., Bates, D., Fleming, A., Ninness, B., Moheimani, S.: Application of MPC to an active structure using sampling rates up to 25kHz. In: 44th IEEE Conference on Decision and Control and European Control Conference ECC’05, Seville (2005)
[58] Wolfe, P, The simplex method for quadratic programming, Econometrica, 27, 382-398, (1959) · Zbl 0103.37603
[59] Wright, S.: Primal-Dual Interior-Point Methods. SIAM Publications, Philadelphia (1997) · Zbl 0863.65031
[60] Wunderling, R.: Paralleler und Objektorientierter Simplex-Algorithmus. PhD thesis, Konrad-Zuse-Zentrum Berlin (1996) · Zbl 0871.65048
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